Newark Mayor Cory Booker announces his plans to run for the U.S. Senate seat during a news conference in Newark.

EDGEWATER, N.J.—Newark Mayor Cory Booker pledged to tackle childhood poverty in the first policy release of his U.S. Senate campaign Monday, while he called the endorsement of challenger Rep. Frank Pallone by the family of the late Sen. Frank Lautenberg “not surprising.”

Mr. Pallone picked up the endorsement of Lautenberg’s family Monday for the senator’s former seat. Four Democrats and two Republicans are running in the heated Aug. 13 primary for the special election.

“Frank Pallone worked with Frank Lautenberg for many years. He understands what it takes to take on and defeat Republicans and the special interests that attack the well being of working families,” said the family in a statement.

Mr. Booker was never named in the release, but it stated that the replacement for Mr. Lautenberg should be a “workhorse, not a showhorse.” The Lautenberg family endorsement came from the late senator’s wife, Bonnie, and his children, Nan, Ellen, Lisa and Josh, a Pallone campaign spokesman said.

“Frank Pallone knows that gimmicks and celebrity status won’t get you very far in the real battles that Democrats face in the future,” reads the statement. “When New Jersey Democrats examine the Senate candidates closely, they may be surprised to find out that not all of them share core Democratic values or loyalty to the party.”

Mr. Pallone, a Monmouth County Democrat, has been critical of Mr. Booker’s record in Newark and his relationship with Republican Gov. Chris Christie.

Speaking at a Bergen County school Monday morning, Mr. Booker brushed aside the endorsement, saying that it was logical given that Messrs. Lautenberg and Pallone worked together for many years. “We expected it,” he said.

If elected, Mr. Booker said he would work to increase the federal minimum wage to $10.10 per hour over the next two years, expand the Earned Income Tax Credit for married couples and provide more support for poor, first-time mothers.

He said that resistance to increase the minimum wage on a state and federal level “has been very misguided.” Mr. Booker said he’d create a coalition to fight for a raise in Congress, but didn’t elaborate on who it would include.

He called child poverty “a national epidemic” in America.

“This isn’t just an urban problem or suburban problem or rural problem. This is an American problem,” Mr. Booker said.